The type-I seesaw mechanism explains why neutrino masses are small — a heavy sterile Majorana mass suppresses the active-neutrino masses — but it cannot predict their actual values: the Yukawa couplings and the right-handed Majorana masses are free parameters, so the absolute neutrino mass scale is fitted rather than derived. This problem asks what dynamics, if any, fixes that scale: can the sub-eV active-neutrino masses (and the associated heavy sterile scale) be derived as predictions from a principled mechanism rather than imported through unexplained boundary assumptions? The question is stated without restriction to a particular model class.

Background

Neutrinos have mass, but their masses are extraordinarily small compared with all other fermion masses: oscillation experiments have measured the two mass-squared splittings, and cosmological bounds on the sum of neutrino masses confine the lightest mass to below roughly 0.1 eV, about seven orders of magnitude below the electron mass. In the Standard Model augmented by right-handed neutrinos, a Dirac mass term couples each left-handed neutrino to a sterile right-handed neutrino νR\nu_R through a Yukawa coupling (a dimensionless coupling like the electron's Yukawa coupling) and the electroweak Higgs vacuum expectation value; getting masses below 0.1 eV this way would require Yukawa couplings below 10−1110^{-11}, which is widely considered unnatural. The type-I seesaw mechanism instead assigns νR\nu_R a Majorana mass (a mass term coupling a fermion to its charge conjugate, which violates lepton number — the conserved quantum number carried by leptons, with neutrinos carrying +1+1). For a Majorana mass MM not tied to electroweak symmetry breaking, MM may be very large, and diagonalizing the full mass matrix gives active neutrinos a mass of order mD2/Mm_D^2/M, naturally small when MM is large. This explains smallness, but as Ibarra, Strobl and Toma put it in the source paper, the seesaw model 'cannot predict the concrete value of the neutrino masses, as the Yukawa couplings and the right-handed Majorana masses are a priori undetermined' — every number in the observed spectrum is a free input. Two routes toward removing this freedom motivate the present question. First, Ibarra, Strobl and Toma (Phys. Rev. Lett. 122, 081803 (2019)) studied a scenario in which lepton number is broken only at the Planck scale, possibly by gravitational operators: with one sterile Majorana mass of order the Planck mass MPM_\mathrm{P} and hierarchical, order-one Yukawa couplings, a two-loop radiative mechanism generates the lighter sterile Majorana mass, and one active-neutrino mass eigenvalue comes out at O(0.1)\mathcal{O}(0.1) eV fairly independently of the remaining free lighter-sector parameters — the two lighter active masses are not predicted, and reproducing the observed splitting ratio requires a small Yukawa eigenvalue y2∼10−4−10−1y_2 \sim 10^{-4}-10^{-1}. However, the required hierarchical boundary spectrum at the Planck scale is, in the authors' words, a purely phenomenological assumption — the freedom has been traded for a different unexplained boundary condition, not derived. Second, Beneš, Hošek and Smetana (JHEP 03 (2021) 224) proposed gauging a strong flavor symmetry SU(3)fSU(3)_f ('quantum flavor dynamics') under which the three νR\nu_R form a triplet required by anomaly cancellation: no elementary Majorana mass terms are imposed, and truncated Schwinger–Dyson equations (self-consistent integral equations for fermion self-energies in strongly coupled gauge theory) dynamically generate sterile Majorana masses of order the flavor scale while exponentially suppressing Dirac masses — in principle deriving both scales. But the nonperturbative spectra are undetermined, and the authors report the resulting mass splittings as unrealistic at the present exploratory stage. Reviews at field level (Volkas, arXiv:2409.09992) list identifying the actual Lagrangian responsible for neutrino masses — 'the Lagrangian to put into textbooks' — as a distant, open goal. The problem is thus: what dynamics, if any, determines the absolute neutrino mass scale, so that the observed value is a prediction rather than a fit?

Problem Statement

Identify the dynamics — or establish whether none is needed or none can work — that fixes the absolute scale of neutrino masses, so that the observed sub-eV active-neutrino masses (and the associated sterile-sector Majorana scale, if any) are predicted rather than fitted. In the type-I seesaw the light masses mν∼mD2/Mm_\nu \sim m_D^2/M inherit free Yukawa couplings and a free right-handed Majorana mass MM; the question asks for a principled mechanism that determines these inputs, or the spectrum directly, without replacing them by equally unexplained boundary assumptions. A mechanism counts as fixing the scale only if it yields the masses from a fixed input set that contains at most one free dimensionful input — the single input scale of the mechanism — with all remaining inputs dimensionless; introducing any further free dimensionful parameter, whatever its origin, relocates the input rather than deriving the scale. No restriction to a particular model class is intended. The task is resolved by either (a) a concrete mechanism that fixes the absolute scale or (b) a no-go result of general scope; an argument that no dynamics is needed resolves the task only if it supplies an actual derivation of the observed scale, and a demonstration that a single proposed mechanism's boundary assumptions are themselves derivable is a constituent step toward (a) rather than a full resolution.

References

  1. Alejandro Ibarra, Patrick Strobl, Takashi Toma, 'Neutrino masses from Planck-scale lepton number breaking', Phys. Rev. Lett. 122 (2019) 081803, DOI 10.1103/PhysRevLett.122.081803, arXiv:1802.09997, https://arxiv.org/abs/1802.09997
  2. Petr Beneš, Jiří Hošek, Adam Smetana, 'Electroweak SU(2)_L × U(1)_Y model with strong spontaneously fermion-mass-generating gauge dynamics', JHEP 03 (2021) 224, DOI 10.1007/JHEP03(2021)224, arXiv:2011.03312, https://arxiv.org/abs/2011.03312
  3. Alejandro Ibarra, Patrick Strobl, Takashi Toma, 'Two-loop renormalization group equations for right-handed neutrino masses and phenomenological implications', Phys. Rev. D 102 (2020) 055011, DOI 10.1103/PhysRevD.102.055011, arXiv:2006.13584, https://arxiv.org/abs/2006.13584
  4. Cesar Bonilla, Johannes Herms, Alejandro Ibarra, Patrick Strobl, 'Neutrino parameters in the Planck-scale lepton number breaking scenario with extended scalar sectors', Phys. Rev. D 103 (2021) 035010, DOI 10.1103/PhysRevD.103.035010, arXiv:2012.11567, https://arxiv.org/abs/2012.11567
  5. Stefan Antusch, Miguel Levy, Salvador Centelles Chuliá, 'Gravity-assisted neutrino masses', Phys. Lett. B 876 (2026) 140410, DOI 10.1016/j.physletb.2026.140410, arXiv:2512.16862, https://arxiv.org/abs/2512.16862
  6. Ilaria Brivio, Jim Talbert, Michael Trott, 'No-go limitations on UV completions of the Neutrino Option', Phys. Rev. D 103 (2021) 015012, DOI 10.1103/PhysRevD.103.015012, arXiv:2010.15428, https://arxiv.org/abs/2010.15428
  7. Raymond R. Volkas, 'Neutrino theory: open questions and future opportunities', PoS(ICHEP2024) 025, DOI 10.22323/1.476.0025, arXiv:2409.09992, https://arxiv.org/abs/2409.09992